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Spectral Theory and Nonlinear Functional Analysis





ISBN 9781584882497
Published March 28, 2001 by Chapman and Hall/CRC
280 Pages

 
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Book Description

This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure of the set of zeroes of a general class of nonlinear operators. It features the construction of an optimal algebraic/analytic invariant for calculating the Leray-Schauder degree, new methods for solving nonlinear equations in Banach spaces, and general properties of components of solutions sets presented with minimal use of topological tools. The author also gives several applications of the abstract theory to reaction diffusion equations and systems.

The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers. Appealing to a broad audience, Spectral Theory and Nonlinear Functional Analysis contains many important contributions to linear algebra, linear and nonlinear functional analysis, and topology and opens the door for further advances.

Table of Contents

INTRODUCTION
General Assumptions and Basic Concepts
Some New Results
Historical Remarks
BIFURCATION FROM SIMPLE EIGENVALUES
Simple Eigenvalues and Transversality
The Theorem of M.G. Crandall and P.H. Rabinowitz
Local Bifurcation Diagrams
The Exchange Stability Principle
Applications
FIRST GENERAL BIFURCATION RESULTS
Lyapunov-Schmidt Reductions
The theorem of J. Ize
The Global Alternative of P.H. Rabinowitz
The Theorem of D. Westreich
THE ALGEBRAIC MULTIPLICITY
Motivating the Concept of Transversality
Transversal Eigenvalues
Algebraic Eigenvalues
Analytic Families
Simple Degenerate Eigenvalues
FUNDAMENTAL PROPERTIES OF THE MULTIPLICITY
The Multiplicity of R.J. Magnus
Relations between c and m
The Fundamental Theorem
The Classical Algebraic Multiplicity
Finite Dimensional Characterizations
The Parity of the Crossing Number
GLOBAL BIFURCATION THEORY
Preliminaries
Local Bifurcation
Global Behavior of the Bounded Components
Unilateral Global Bifurcation
Unilateral Bifurcation for Positive Operators
APPLICATIONS
Positive Solutions o Semilinear Elliptic Problems
Coexistence States for Elliptic Systems
Examples
A Further Application
REFERENCES
INDEX

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Reviews

"The book is both an excellent introduction to some novel ideas about nonlinear eigenvalue problems and an exposition of a range of earlier results scattered in different papers and expounded in book form for the first time here."
- Mathematical Reviews, Issue 2002

"This is a nice introductory text about classical functional analysis…" This book will be interesting and useful for many mathematicians, scientists, graduate and undergraduate students."
-Mathematical Reviews